Some Comments about Chapter 3 of Hollander & Wolfe
Section 3.1
I get annoyed with H&W. The "main part" of each section is
"cookbookish" and then details are given in the Comments at the
end of the section. But the comments tend to address selective details,
and I feel as though a nice description of the procedures, with good
motivation and discussion, is omitted. Perhaps our discussion in the
classroom will fill in some gaps. Meanwhile, I'll offer some comments
about the text below.
- p. 36, Assumptions
- While each individual in the population may have a unique
distribution for the Y-X difference, if the sample of
Zi is obtained by randomly drawing from the
population, we can think of the
Zi
as being identically distributed (having as their common distribution,
the mixture distribution obtained by giving weight 1/N, where
N is the population size, to each of the N
Y-X difference
distributions). I suppose a good thing about the test is that if the
treatment has a constant effect on each member of the population, one
doesn't really need a random sample --- all that is needed, besides
independence, is that
after an individual from the population is selected, the measured
difference will be governed by a distribution which is symmetric about
theta. One of the claims sometimes made about nonparametric
tests is that random samples are not really needed for the tests to
be meaningful. To explore this notion further, suppose that the
treatment does nothing, and that all of the difference distributions are
then symmetric about 0. As long as we have independence, the
signed-rank test statistic's distribution should follow it's usual null
distribution, no matter how the individuals (subjects) were selected.
But from my point of view, if a rejection of the null hypothesis of no
treatment effect is obtained, all I have is evidence of some sort of a
treatment effect. Not being a believer (for most settings) in a
constant treatment effect, if the data can be viewed as a random sample,
I could use the data to make inferences about the nature of the
treatment effect for the population which the random
sample represents. (E.g., I could estimate the median change and/or the
mean change, or perhaps the proportion of individuals in the population
for which the change will be at least as large as 5.) But if I cannot
view the data as a random sample, I would be hesitant to use it to make
an inference about a larger population --- all I would feel confident in
believing is that I have strong evidence that the treatment has some
effect on at least a portion of the population. If I can believe in a
constant treatment effect (which I might be able to do if my
observations are just repeated measurements on the same treated unit,
and the variation is just due to measurement error, which has the same
distribution before and after the treatment), then I suppose I can go
onwards from a rejection of the null hypothesis and use the methods of
Secctions 3.2 & 3.3 to estimate the value of the treatment effect,
theta. (Perhaps I should discuss all of this "philosophical
stuff" in class. There are some subtle points involved, and it may take
you a while to get comfortable with everything.)
- p. 37, a & b
- Note that talpha is an integer. For each
n, only certain
values are possible for alpha (not necessarily 0.05, 0.025, 0.01,
and 0.005). If, instead of using a test having a preset size, one just
wants the p-value, then for an upper-tail test you can look up the value
of t+ in the x column of Table A.4 for the
right value of n, and the p-value is the probability given in the
next column over. For a lower-tail test, you look up
n(n+1)/2 - t+ in the x column.
- p. 40
- Here are some comments about Example 3.1.
- If one wanted to round the p-value to two significant digits
(which, even though it's an exact p-value, can still benefit from
rounding since it's not all that important whether a p-value is 0.0195
or 0.0196), we don't know how to round 0.0195 (i.e., whether it should
be 0.019 or 0.020). I don't like the default in StatXact, which is to
round the p-value to the nearest ten thousandth. Sometimes one waits a
while to get a p-value, and then StatXact prints out 0.0000. To avoid
being disappointed if this were to happen, I suggest that before
doing any tests, click on Options on the top bar of the main
window of StatXact, and then select Global. For Display of
Numeric Output, select Exponential, which will result in
p-values given in scientific notation, always displaying some
significant digits (and never giving all zeros). Before closing, go
down to the bottom and put a check beside of Save Gloabal Parameters
Permanently (which will keep the scientific notation while you
continue to use StatXact for additional tests during your
session, but unfortunately will not be remembered after you shut down
StatXact, and so you have to change it each time you start up
StatXact). Now when the exact signed-rank test is run, one can
see that the one-sided p-value rounds to 0.019531, which means that if
we want two significant digits, it should be 0.020, and not 0.019.
- As H&W point out, Minitab uses a normal approximation with a
continuity correction. So the 5 in T* becomes 5.5,
and
T* becomes -2.014, and one obtains an approximate
p-value of about 0.022. In this case, the continuity correction made
the approximation worse. I find that this is sometimes the case when
one should not even be using an approximation. That is, if n is real
small, the approximation w/o the cont. corr. may be better than the
approx. w/ the cont. corr., but of course an exact p-value is better
than any approximation, and there is no good reason not to report an
exact p-value. I find that the cont. corr. can also make the approx.
worse when the p-value is real small, but in most cases that I've
investigated, with a sample size larger than 20, the cont. corr. made
the normal approx. better.
- An exact permutation test yields a one-tailed p-value of about
0.014, which is smaller than the p-value from the signed-rank test.
I'll go over the permutation test in class since H&W doesn't cover it.
- pp. 41-42
- Here are some comments about Example 3.2.
- In the book, a conservative approach (which is good if you don't
have StatXact) is used to obtain the "exact" p-value of 0.039 (which is
the p-value corresponding to a test statistic value of 62, instead of
the value of 62.5). StatXact can be used to arrive at an exact p-value
based on the midrank scheme. (The exact method using midranks is
explained in Comment 11 on pp. 46-48.) Below I will compare
various p-values, obtained using different methods.
- 0.0334 (StatXact, using midranks)
- 0.039 (conservative scheme using table, looking up 62)
- 0.0386 (same as above, if we had a more accurate table)
- 0.0326 (normal approx. using 62.5, variance corr., but no
cont. corr.)
- 0.036 (Minitab's normal approx. using 62.5,
w/ a 1/2 cont. corr.)
This illustrates that if one didn't have StatXact, the normal
approximation based on the value of 62.5 comes closer to the exact value
of about 0.033 from StatXact than does the conservative value of 0.039
obtained from the table.
- I suspect the set of 12 differences is not a random sample. Having
to find pairs of workers with very similar characteristics would further
complicate the difficult task of drawing a random sample.
- p. 44, Comment 7
- The key to obtaining the null distribution of the signed-rank
statistic (and easily obtaining the null mean and variance) is to use
the fact that the null distribution of the test statistic is the same as
the sum of the Vi, with the
Vi being independent, having the distribution given near
the bottom of the page. I'll explain the equivalence in class. (I'll
start with an example of a randomized experiment, and then provide an
argument for the more general case.)
- p. 45, Comment 7
- The special case of the central limit theorem which pertains to iid
random variables cannot be used to establish the asymptotic normality of
the signed-rank statistic, since the statistic cannot be viewed as a sum
of iid random variables. A more general version of the central limit
theorem, like would be covered in a Ph.D. level probability class, is
needed. Because of this, I won't establish the asymptotic normality in
class.
- p. 45, Comment 8
- H&W point out that the symmetry is verified for the n = 3
case in the book, but it's easy to see why the null distribution is
symmetric for any value of n. The key is to note that there is a
1:1 correspondence between sets of integer ranks that sum to x,
and sets of integer ranks that sum to n(n+1)/2 - x. (Ask me
about this in class if you don't understand why the null sampling
distribution is symmetric.)
- p. 46, Comment 9
- I'll discuss dealing with values of 0 in class. (With a randomized
experiment, it's clear that they should be ignored (as H&W suggests in
general). The StatXact manual also suggests an alternative
strategy.)
- p. 49, Comment 13
- Note that H&W use beta for the probability of a type II
error, instead of for power, and so 1 - beta is the power.
Section 3.2
I'll describe the general scheme for Hodges-Lehmann estimators associated
with nonparametric tests in class. (The estimator covered in this
section is a special case of a class of estimators.)
Section 3.3
H&W doesn't give a lot of explanation for the confidence interval
covered by this section. (They don't show why it would have the stated
coverage probability.) If there is time, I may say a little more
about this interval during the 3rd lecture, but I won't take time
to do so during the 2nd lecture --- I think other topics are more
important to discuss.
- p. 58, Comment 23
- Yet another estimator is suggested. Since no indication of when
this may be a good estimator to use is supplied, I think it's safe to
ignore/forget this estimator.
Section 3.4
I'll give my comments below, starting with one pertaining to the top
portion of p. 60, which is actually not part of Sec. 3.4 (and doesn't
seem to be part of any section --- H&W tend to have little introduction
portions like this, with the top portion of p. 60 serving as an
introduction of sorts to Sections 3.4, 3.5, & 3.6).
- p. 60, top portion
- At first, the fact that B2 doesn't require that all of the
differences have the same distribution may seem somewhat liberating.
But, since it would be very odd for a treatment to have a variable
effect, yet have the median difference be the same for each individual,
it seems like B2 would only hold if the treatment had the exact
same effect --- causing a constant change, theta, in all
individuals --- and the variability was just random measurement error
(or perhaps some sort of natural fluctuation of the phenomenon being
measured). From such a viewpoint, the fact that B2 doesn't
require that all differences have the same distribution seems to impose
a rather severe restriction on the nature of the treatment effect, and
is far from being liberating. I would prefer to be able to view the
observed differences as being a random sample (from a common
distribution), since in such a case theta can represent the
median difference caused by the treatment (when applied to everyone in
the population), and we don't have to assume that the treatment had
the same effect on all individuals.
- p. 60, (3.39)
- We can view (3.39) as being of the same general form as (3.3), each
being a sum of nonrandom
scores multiplied by iid indicator random
variables. For the signed-rank test, the scores are the integers, 1, 2,
3, ..., n, and for the sign test, each score has the same value
of 1. To create another nonparametric test, we only need to specify a
different set of scores. I'll do this in class when I introduce the
one-sample / paired-samples normal scores test.
(Annoyingly, neither H&W nor
StatXact include the one-sample / paired-samples normal scores
test, although both include the two-sample version, aka Van der
Waerden's test.) The one-sample / paired-samples permutation test can
also be written in the same general form as (3.3), only the scores
differ from time to time, since they are the absolute values of the
observations. (If the null hypothesis value isn't 0, one first subtracts the
null hypothesis value from each observation and then takes absolute
values.)
- p. 61, (3.45)
- For the normal approximation of the sign test, a continuity
correction typically improves the accuracy.
- p. 62, Ties
- With data from a matched-pairs randomized experiment, it definitely
makes sense to ignore the zero values when testing for a treatment
effect. (I'll explain this in class.)
- p. 64, Comment 26
- The difference between A2 and B2 is that B2
doesn't specify that the distributions are symmetric. B2' allows
for the possibility that the distributions are not continuous (but does
require that P(Zi = theta0) = 0,
since it must be that
P(Zi < theta0) =
P(Zi > theta0) = 0.5).
- pp. 68-69 (bottom of p. 68, top of p. 69)
- If a continuity correction is used, the approximate power is 0.5,
which is pretty close to the exact power. (Note: If the cont. corr.
isn't used, someone could do the computation another way and arrive at
0.6425, instead of 0.3575, as the approximate power. So not only does
the cont. corr. improve the accuracy here, it also gives a unique
value for the approximate power, which is not the case if the correction
isn't used.)
- p. 69, (3.56)
- I might derive (3.56) in class.
Section 3.5
- p. 72
- Even though the sample median has a connection with the sign test
(it's the Hodges-Lehmann est'r associated with the sign test), I think
it's important to keep in mind that the sample median is not a
particularly good estimator of the medain for most distributions. The
Harrell-Davis estimator is typically a better choice (although it is not
resistant to gross outliers, like the sample median is, and so one must
watch out for them). Also, in many small
sample size situations, various trimmed means and M-estimators work
better than the sample median (and sometimes the sample mean is better),
even though these alternatives aren't necessarily consistent estimators
of the distribution median unless the distribution is symmetric.
- pp. 72-73, Comment 38
- One can call the sample median the Hodges-Lehmann estimator
associated with the sign test, but since there are so many different
Hodges-Lehmann estimators (not all associated with nonparametric
procedures --- there is a H-L estimator for the error term variance in
an ANOVA model), I think it's better to simply refer to the estimator
dealt with in this section as the sample median.
- p. 73, Comment 43
- My guess is that in most situations, many other estimators will
outperform a quasimedian. I doubt that I would ever choose to use a
quasimedian in practice (unless I was trying to estimate the median of a
uniform diestribution, with both endpoints of the support being unknown,
in whcih case I would use the average of the sample minimum and the
sample maximum, which is an extreme example of a quasimedian).
- pp. 73-74, Comment 44
- Estimators having the form given in (3.61) are called
L-estimators. Sample medians, the sample mean, trimmed means,
and quasimedians are all L-estimators.
- p. 74, Comment 45
- It is true that for some distributions, with an odd sample size,
one is better off deleting
the last observation in the sample (not the ordered sample), and
computing the sample median from a sample having an even sample size.
I believe that this is true for a uniform distribution, but not
all distributions. (Indeed, in Hodges and Lehmann's 1967 paper,
in their TABLE 1.4, there is an example which indicates that it isn't
always true for normal distibutions, since the variance of the sample
median for the n = 19 case is smaller than the variance of
the sample median for the n = 18 case.)
- p. 74, Comment 46
- I've never seen this estimator for the asymptotic standard error of
the sample median given anywhere else, although I've seen several
alternatives suggested in various places. None of the alternatives that
I've investigated work well when the sample size is smallish, and I
suspect that the same is true for this estimator (after all, it is
referred to as an estimator of the asymptotic standard deviation).
Section 3.6
Note that only certain coverages probabilities are possible for exact
intervals (with the choices
depending on the sample size, and not necessarily including 0.9, 0.95,
and 0.99). Minitab's sint command will produce an
approximate interval having any approximate coverage probability,
using a nonlinear interpolation method. Pages 87-88 of Rand Wilcox's
Introduction to Robust Estimation and Hypothesis Testing (Academic
Press, 1997) describes such a method, provides references, and indicates
that some researchers have supported the use of the method. But when
n is rather small, I prefer to go with an exact interval, even
though I may have to use a nonstandard coverage probability.
- p. 78, Comment 52
- The estimators described here are the same as the quasimedians
described in Comment 43.
- p. 78, Comment 54
- Note that if the distribution is discrete, one can use the same
formulas (as in the continuous case) to obtain exact confidence interval
endpoints, only in the discrete case the interval endpoints are part of
the interval --- that is, we need to write the interval as a closed
interval instead of an open interval. If the phenomenon being measured
has a continous distribution, but limited precision in measuring, or
excessive rounding, creates tied values, then one should perhaps express
a confidence interval as a closed interval. For example, if
z(L) is the lower confidence bound, and
z(U) is the lower confidence bound, if
z(L+1) equals
z(L), or
z(U-1) equals
z(U), then I would write the interval as
[z(L),
z(U)], and not
(z(L),
z(U)).
Section 3.7
There isn't a lot new in this section (given that one has gone through
Sections 3.4, 3.5, and 3.6). During the 3rd lecture, I'll extend
the coverage to deal with making inferences about quantiles other than
the median.
- p. 80
- Example 3.8 seems a bit silly --- I don't think it's
pertinent to do a two-sided test using 81.3035 as the null hypothesis
value, since 81.3035 seems to be an estimate obtained from previously
obtained data, and so it seems to be more of a two-sample problem than a
one-sample problem.
- p. 81
- The sentence before the Comments section points out that the
"data provides an example in which the populations of the Z
observations are probably not the same." I guess they mean that since the
experimental conditions were not the same from time to time, the
distribution of the measurement error should not be assumed to be
the same. This seems okay to think, but then why should we think that
each distribution is symmetric about the same value? (Why should we
assume that the error distribution is symmetric? Why should we even
think that the expected value of the measurements is the value that
we're trying to estimate? One of the nice things about doing a test on
the differences of paired data is that it can be hoped that the
differencing operation cancels out any measurement bias (and also
insures symmetry if there is no treatment effect.)
Section 3.8
- p. 84
- See p. 61 if you have forgotten about the
balpha,1/2 notation.
- p. 85
- The approximate confidence interval example right before the
Comments section is silly. In my opinion, the beauty of the
approximate confidence interval formula is that one can use it to obtain
an approximate 95% or 99% confidence interval when an exact interval is
not possible.
- p. 85, Comment 57
- I'll cover making inferences about other quantiles in class.
Section 3.9
As far as I know,
the test described in this section isn't included on StatXact or any
other major statistical software package. Since it's a pain to perform unless
n is rather small, and since the test has little power to detect
asymmetry unless n is not small, I can't imagine that this
test gets a lot of use (and in fact, I'll guess that it get's
practically no use at all). If a distribution is skewed, it's usually
the case that the skewness can be detected with graphical methods unless
n is rather small. But, I suppose that when n is large enough for the test
to have decent power, it can be used to partially
confirm the apparent
skewness. However, I think that cases for which one would like to have
evidence of symmetry are more common than cases where one seeks evidence
of skewness, and the test is of little value for providing evidence of
symmetry, since a failure to reject the null hypothesis could be due to
low power to detect skewness.
My guess is that while most books on nonparametric statistics omit this
test, it is included here because one of the authors is partially
credited with its development. I'm not going to emphasize this test
because we don't have an easy way to perform it, and it just doesn't
seem that useful to me.
The first published article about this type of "triples test" for
asymmetry appeared in 1978, and was written by Davis and Quade. The
article about this test by Randles, Fligner, Policello, and Wolfe
appeared in 1980, but was originally submitted to the journal in July of
1977. So it may be the case that both teams of authors developed the
same test at about the same time, with the Davis and Quade article
appearing first in a journal that typically had a shorter time lag
between article submission and article publication. It can be
noted that there was a 23 month gap between the time the Randles et al.
article was first submitted and when the revision (that eventually
appeared in
print) was submitted. I'll guess that most of that time was taken by
the journal's editors and the article's referees to process the original
submission. Since the 1980 article contained additional information
about the test, I have no problem with giving the two sets of authors
equal billing.
It can be noted that the test described in Sec. 3.9 apparently can be
anticonservative (i.e., if the nominal size of the test is 0.05, the
actual probability of a type I error may exceed 0.05). Randles et al.
recommend using t critical values (using n df) instead of
standard normal critical values to help curb the anticonservative
behavior. This ploy helps (using the larger t critical values
will result in fewer type I errors), but the test may still be
anticonservative for certain parent distributions of the data. Another
test for asymmetry is based on the asymptotic normality of the sample
skewness. A Monte Carlo study done by Randles et al. indicates that
this competitor test has less problems with anticonservativeness, but also
rejects less than the "triples test" when the alternative hypothesis is
true (i.e., it has lower power). So it seems to come down to a choice
of using a more powerful, but less accurate (with respect to respecting
the stated level of the test) test, and a test which seems to misbehave
less under the null hypothesis, but has lower power. Some would
argue strongly that the test that better respects the nominal type I
error rate should be chosen, while others would choose the more powerful
test as long as they felt that the actual type I error rate wasn't too
inflated above the nominal level of the test. This second viewpoint
wouldn't be so bad if we could characterize the types of situations in
which the "triples test" badly misbehaves, be able to identify when those
situations arise in practice, and avoid using the "triples test" in such
cases.
Below I'll summarize some results presented in the Randles et
al. paper. T designates the "triples test" based on standard
normal critical values,
T* designates the "triples test" based on
tn critical values, and S designates the
competitor test based on the asymptotic normality of the sample
skewness.
The first two tables below show estimated type I error rates for the three
tests, for six different symmetric parent distributions.
The kurtosis of the distributions increase as one goes from the
1st distribution to the 5th distribution (I may add the
exact kurtosis values later), and the kurtosis does not exist (which
suggests really heavy tails) for the 6th distribution.
| n = 20 | T |
T* |
S |
| distribution 1 |
0.071 |
0.055 |
0.038 |
| distribution 2 |
0.045 |
0.034 |
0.023 |
| distribution 3 |
0.064 |
0.047 |
0.037 |
| distribution 4 |
0.067 |
0.053 |
0.041 |
| distribution 5 |
0.068 |
0.056 |
0.041 |
| distribution 6 |
0.103 |
0.083 |
0.056 |
| n = 30 | T |
T* |
S |
| distribution 1 |
0.079 |
0.065 |
0.040 |
| distribution 2 |
0.050 |
0.037 |
0.023 |
| distribution 3 |
0.058 |
0.048 |
0.032 |
| distribution 4 |
0.061 |
0.049 |
0.032 |
| distribution 5 |
0.065 |
0.056 |
0.035 |
| distribution 6 |
0.089 |
0.080 |
0.020 |
T* should be clearly preferred to T, since both
can be anticonservative, but T*'s performance isn't as
bad. One can note that while
T* can have an inflated type I error rate, it isn't
badly inflated except for the extremely heavy-tailed parent
distribution, and for this distribution S also has an inflated
type I error rate when n = 20. It might be worthwhile to adjust the "triples test"
a bit more, perhaps using t critical values with n-1 or
n-2 df (since the slightly larger critical values will dampen the
anticonservativeness). Another idea would be to use bootstrapping to
improve the test based on the sample skewness --- if it could be made to
be less conservative in cases for which it is conservative, then it's
power should be improved in those cases.
The next two tables show estimated powers (against a particular
alternative) for the two most accurate
tests, for fourteen different skewed parent distributions.
The skewness is given for eight of the distributions.
| n = 20 | skewness |
T* |
S |
| distribution 7 |
0.50 |
0.222 |
0.147 |
| distribution 8 |
1.50 |
0.625 |
0.301 |
| distribution 9 |
0.90 |
0.190 |
0.125 |
| distribution 10 |
1.50 |
0.286 |
0.179 |
| distribution 11 |
0.80 |
0.060 |
0.044 |
| distribution 12 |
2.00 |
0.129 |
0.090 |
| distribution 13 |
3.16 |
0.769 |
0.323 |
| distribution 14 |
3.88 |
0.793 |
0.324 |
| distribution 15 |
|
0.248 |
0.192 |
| distribution 16 |
|
0.446 |
0.384 |
| distribution 17 |
|
0.219 |
0.153 |
| distribution 18 |
|
0.624 |
0.367 |
| distribution 19 |
|
0.140 |
0.088 |
| distribution 20 |
|
0.304 |
0.100 |
| n = 30 | skewness |
T* |
S |
| distribution 7 |
0.50 |
0.341 |
0.262 |
| distribution 8 |
1.50 |
0.817 |
0.452 |
| distribution 9 |
0.90 |
0.325 |
0.175 |
| distribution 10 |
1.50 |
0.478 |
0.222 |
| distribution 11 |
0.80 |
0.072 |
0.038 |
| distribution 12 |
2.00 |
0.209 |
0.089 |
| distribution 13 |
3.16 |
0.924 |
0.380 |
| distribution 14 |
3.88 |
0.940 |
0.357 |
| distribution 15 |
|
0.345 |
0.287 |
| distribution 16 |
|
0.606 |
0.579 |
| distribution 17 |
|
0.393 |
0.232 |
| distribution 18 |
|
0.846 |
0.562 |
| distribution 19 |
|
0.164 |
0.026 |
| distribution 20 |
|
0.396 |
0.048 |
One can see that the power values are generally considerably higher for
the the "triples test", and so it would be nice to find a way to correct
it's anticonservativeness problem. Note that for the smallish sample
sizes considered in the Monte Carlo study, the power can be rather low
when the skewness is less than 3 (and so while it's perhaps the better
of the two tests, it can have disappointingly low power, since a
skewness of 3 is fairly large in my opinion, and one might think that
smaller skewnesses should be detected with higher probability than some
of the low powers observed).
Below are some more comments.
- p. 93, Comment 64
- This scheme seems a lot easier than the one described on p. 88.
One just needs to compare the average of the minimum and maximum of the
two values to the middle value.
- p. 93, Comment 65
- A more theoretical course than STAT 657 would spend a bit of time
on the general theory of U-statistics. But such a course would
be more appropriate for Ph.D. students who want to do research on
U-statistics, and develop a new test procedure. The focus of
STAT 657 is on the correct application of exisiting test procedures.
Note that here
U-statistics refer to a large class of statistics, and not what
is sometimes referred to as the Mann-Whitney
U statistic, which is just one member of the more general class
of U-statistics.
Section 3.10
As far as I know,
the test described in this section isn't included on StatXact or any
other major statistical software package. Since it's a pain to perform unless
n is rather small,
I can't imagine that this
test gets a lot of use (and in fact, I'll guess that it get's
practically no use at all).
The null hypothesis under consideration in this section implies that the
distribution of Xi - Yi is symmetric about 0,
and we can shoot down that hypothesis with easier-to-perform tests such
as the
signed-rank test, the sign test, the normal scores test, and the permutation
test.
For example, using the data from Problem 3.113 on p. 104 of the text
(which is the data in Table 3.3 on p. 50), one gets a p-value of 0.125
using the test from Sec. 3.10, but one can get that same p-value much
easier using the sign test, and one can get the smaller p-value of about
0.1094 using the signed-rank test. (For the data in Example 3.11 on p.
97, one can get smaller p-values using the sign test, the signed-rank
test, and the normal scores test, than one can get with the test from
Sec. 3.10. But none of the p-values are very small, and so perhaps
the differences in their values don't mean a lot.)
Section 3.11
I'll work through an example in class to show how to obtain results like
those given by (3.116) on p. 104 and (3.118) on p. 105. (I will assign
a homework problem or two that will instruct you to find similar results
for other distributions.)